We present a systematic inelastic neutron scattering and neutron diffraction study on the magnetic structure of the quasi-one-dimensional spin-1/2 magnet SrCo₂V₂O₈, where the interchain coupling in the Néel-type antiferromagnetic ground state breaks the static spin lattice into two independent domains. At zero magnetic field, we have observed two new spin excitations with small spectral weights inside the gapped region defined by the spinon bound states. In an external magnetic field along the chain axis, the Néel order gets partially destabilized at μ₀H^★ = 2.0 T and completely suppressed at μ₀H_p = 3.9 T, above which a quantum disordered Tomonaga–Luttinger liquid (TLL) prevails. The low-energy spin excitations between μ₀H^★ and μ₀H_p are not homogeneous, containing the dispersionless (or weakly dispersive) spinon bound states excited in the Néel phase and the highly dispersive psinon-antipsinon mode characteristic of a TLL. We propose that the two new modes at zero field are spinon excitations inside the domain walls. Since they have a smaller gap than those excited in the Néel domains, the underlying spin chains enter the TLL state via a local quantum phase transition at μ₀H^★, making the Néel/TLL coexistence a stable configuration until the excitation gap in the Néel domains closes at μ₀H_p.